Let's first recall how to evaluate a logarithm. A logarithm is the inverse of an exponential function. The expression log_b(m) is read as log base b of m, and states that raising b to the n^(th) power yields m.
n=log_b(m) ⇔ b^n=m
We will take each function and rewrite it in exponential form. Then, we will substitute the values into the table to see if they satisfy the function. Let's start with option F.
y=log_(12)x
Let's first rewrite our function in exponential form.
y=log_(12) x ⇔ 1/2^y= x
Now we can substitute the given values of y from our table to see if they produce the corresponding x-values.
For option F, the corresponding x-value to y=-1 is x=2. From the table we see that the corresponding x-value must be x=1/2, so we know option F is incorrect. Let's move on to option G.
y=- log_2x
We can see our function y=- log_2x has the value - 1 in the front. In order to rewrite our function in exponential form, we must first recall the Power Property of Logarithms.
y=nlog_b(m) ⇔ y=log_b(m^n)
Using this rule, we can first rewrite our function as follows.
y=(-1) log_2 x ⇔ y=log_2( x^()darkorange-1)
Now we can rewrite our function in exponential form.
y=log_2( x^()darkorange-1) ⇔ 2^y= x^()darkorange-1
We will substitute the given values of x and y from our table to see if they satisfy the exponential function.
We see that the given values of x and y do not satisfy the function in option G. Let's try option H.
y=log_2x
Let's rewrite our function in exponential form.
y=log_2 x ⇔ 2^y= x
Now we can substitute the given values of y from our table to see if they generate the corresponding x-values.
From our given table, we see that the value x= 12 does in fact correspond to the y-value of - 1. Let's substitute the rest of the y-values from our table into our function, to ensure it generates the corresponding x-values.
y
x=2^y
x
0
x=2^0
x=1
1
x=2^1
x=2
2
x=2^2
x=4
We see that option H generates the table of values given. Just to confirm that the correct answer is H, let's compute the corresponding y-values for option I.
y=(1/2)^x
Substitute the given x-values from the table into y=( 12)^x.
x
y=(1/2)^x
y
1/2
y=(1/2)^(12)
y≈ 0.71
1
y=(1/2)^1
y=1/2
2
y=(1/2)^2
y=1/4
4
y=(1/2)^4
y=1/16
We can see that our x and y-values do not correspond to the ones given. This reconfirms that the function in option H generates the given table.
